MEMS resonant accelerometer having improved electrical characteristics

ABSTRACT

A MEMS resonant accelerometer is disclosed, having: a proof mass coupled to a first anchoring region via a first elastic element so as to be free to move along a sensing axis in response to an external acceleration; and a first resonant element mechanically coupled to the proof mass through the first elastic element so as to be subject to a first axial stress when the proof mass moves along the sensing axis and thus to a first variation of a resonant frequency. The MEMS resonant accelerometer is further provided with a second resonant element mechanically coupled to the proof mass through a second elastic element so as to be subject to a second axial stress when the proof mass moves along the sensing axis, substantially opposite to the first axial stress, and thus to a second variation of a resonant frequency, opposite to the first variation.

BACKGROUND

1. Technical Field

The present disclosure relates to a MEMS (Micro Electro MechanicalSystem) resonant accelerometer having improved electricalcharacteristics.

2. Description of the Related Art

As is known, MEMS accelerometers play an important role in the field ofsensors with applications in various contexts including automotive,vibration monitoring and portable electronics. The large number ofmicro-accelerometers proposed in the literature and nowadays present onthe market can be grouped in three classes, on the basis of the sensingprinciple: capacitive, resonant and piezoresistive. The more commonsurface micromachined accelerometers belong to the first class, but alsoresonant accelerometers have been produced by surface micromachiningtechnology. In this respect, reference may be made to the followingpapers:

-   M. Aikele, K. Bauer, W. Ficker, F. Neubauer, U. Prechtel, J.    Schalk, H. Seidel “Resonant accelerometer with self-test”, Sensors    and Actuators A, 92, 161-167, 2001;-   A. A. Seshia, M. Palaniapan, T. A. Roessig, R. T. Howe, R. W.    Gooch, T. R. Shimert, S. Montague “A vacuum packaged surface    micromachined resonant accelerometer”, JMEMS, 11, 784-793, 2002;-   L. He; Y.-P. Xu; A. Qiu “Folded silicon resonant accelerometer with    temperature compensation”, Sensors 2004. Proceedings of IEEE, 1,    512-515, 24-27 Oct. 2004;-   S. X. P. Su, H. S. Yang, A. M. Agogino “A resonant accelerometer    with two-stage microleverage mechanisms fabricated by SOI-MEMS    technology” Sensors, 5(6), 1214-1223, 2005.

In resonant accelerometers, the external acceleration produces arecordable shift of the resonance frequency of the structure, or of somepart thereof. Resonant sensing, with respect to other sensingprinciples, has the advantage of direct frequency output, high potentialsensitivity and large dynamic range.

Sensitivity of resonant accelerometers is generally defined as thefrequency shift produced by an external acceleration of 1 g. Knownresonant accelerometers obtained through surface micromachiningtypically have sensitivity ranging from 40 Hz/g up to 160 Hz/g, and, atleast some of them, have quite large dimensions.

A conceptual diagram of a linear accelerometer is shown in FIG. 1. Aninertial mass m is attached to a frame by means of a spring of stiffnessk and is subject to damping from the surrounding environment,represented by a damper of coefficient b. When the reference frame issubject to an external acceleration a, the oscillation of the inertialmass is governed by the dynamic equilibrium equation:m{umlaut over (x)}+b{dot over (x)}+kx=ma

If the frequency Ω of the external acceleration is well below resonance,i.e., if Ω<<ω, ω=√{square root over (k/m)} being the frequency of theaccelerometer, the accelerometer response is quasi-static andx(t)≈(m/k)a(t). The external acceleration turns out to be proportionalto the mass displacement and sensing can be done by measuring the massdisplacement, e.g., via the capacity variation as in known capacitiveaccelerometers.

In resonant accelerometers, instead, the input acceleration is detectedin terms of a shift in the resonant frequency of a sensing devicecoupled to the proof mass. The corresponding scheme is represented inFIG. 2, where a resonating beam, shown horizontally, is the abovesensing device.

The operating principle is based on the dependence of the resonantcharacteristic on the axial force which acts on the resonator. Theexternal acceleration a produces a force, F=ma on the inertial mass m.This force produces, in turn, an axial force N in the resonating beam(which is driven in resonance). For a single span beam, frequencyincreases in the case of a tensile load and decreases in the case of acompressive load.

As is known, denoting by f₀, the fundamental frequency of the beamresonating without axial load, the resonant frequency f of the axiallyloaded beam can be expressed as:

$\begin{matrix}{f = {f_{0}\sqrt{1 + {\alpha\frac{{NL}^{2}}{EI}}}}} & (1)\end{matrix}$wherein:

$\begin{matrix}{f_{0} = {\frac{c^{2}}{2\pi\; L^{2}}\sqrt{\frac{EI}{\rho\; A}}}} & (2)\end{matrix}$and L, A and I are the length, the cross area and the inertial moment ofthe resonator, respectively, E is the elastic modulus, and c and α arecoefficients depending on the boundary conditions of the resonator. Thefollowing table shows the values of these coefficients for severalboundary conditions:

c α clamped-free 1.875 0.376 sliding-pinned 1.572 0.405 pinned-pinned3.142 0.101 sliding-sliding 3.142 0.101 clamped-clamped 4.730 0.0246

As a general rule, the external acceleration and resulting force on theresonators produces a variation in the natural frequency of the sameresonators and by measuring this frequency variation it is possible toobtain the value of the external acceleration.

Several accelerometers based on the resonant operating principle havebeen manufactured, through “bulk micromachining” and “surfacemicromachining” technologies. These known accelerometers have differentgeometry (in particular different arrangements of the resonating beamwith respect to the proof or sensing mass) which greatly affect theamplification of the axial force and hence the sensitivity of theresulting sensor.

None of the proposed sensing structures has proven to be fullysatisfactory in terms of the dimensions and electrical characteristicsof the resulting accelerometer sensors. In particular, sensitivitieslimited to the range 10-160 Hz/g have been obtained with the knownsensing structures having comparable size.

BRIEF SUMMARY

One embodiment is a resonant accelerometer, having improved physical andelectrical characteristics. The resonant accelerometer includes a proofmass, a first anchoring region coupled to the substrate, and a firstelastic element coupled to the first anchoring region and the proofmass, the first elastic element configured to allow movement along asensing axis in response to an external acceleration. The resonantaccelerometer also includes a second anchoring region coupled to thesubstrate, a second elastic element coupled to the second anchoringregion and to the proof mass, the second elastic element configured toallow movement along the sensing axis in response to the externalacceleration. A first resonating element is coupled to the proof massvia the first elastic element, the first resonating element having aresonant frequency and configured to generate a first variation of theresonant frequency in response to a first axial stress when the proofmass moves along the sensing axis. A second resonating element iscoupled to the proof mass via the second elastic element, the secondresonating element having the resonant frequency and configured togenerate a second variation of the resonant frequency in response to asecond axial stress when the proof mass moves along the sensing axis,the second axial stress being substantially opposite to the first axialstress, and the second variation being substantially opposite to thefirst variation.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

For a better understanding of the present disclosure, preferredembodiments thereof are now described, purely by way of non-limitingexample and with reference to the attached drawings, wherein:

FIGS. 1, 2 show schematic representations of known accelerometerstructures;

FIGS. 3 a and 3 b show a sensing structure of a resonant accelerometeraccording to an embodiment of the present disclosure;

FIG. 3 c shows the structure of FIGS. 3 a and 3 b, when subject to anexternal acceleration;

FIG. 4 shows sensitivity plots relating to the structure of FIGS. 3 aand 3 b;

FIG. 5 a, shows portions of a known accelerometer structure;

FIG. 5B shows portions of the sensing structure of the accelerometer ofFIGS. 3 a and 3 b;

FIGS. 6 and 7 show plots of physical and electrical quantities relatingto the structure of FIGS. 3 a and 3 b; and

FIGS. 8 a, 8 b are SEM pictures of the structure of FIG. 3 a.

DETAILED DESCRIPTION

FIGS. 3 a, 3 b show a MEMS sensing structure of a resonant uniaxialaccelerometer according to a first embodiment of the present disclosure,denoted as a whole by 1.

The sensing structure 1 includes a proof (or sensing) mass 2, having agenerically square shape (in a main plane of extension xy) and twoprojections 2 a, 2 b, extending from diagonally opposite corners of theproof mass 2 (e.g., from the top right and bottom left corner in FIG. 3a).

The proof mass 2 is suspended by means of two springs 3 a, 3 b which areso configured to restrain its movement to a single uniaxial translation,along axis A (parallel to reference axis y); springs have an elongatedstructure extending in a direction transversal to the axis A (e.g.,orthogonally thereto, parallel to reference axis x). In more detail,springs 3 a, 3 b can be of a single beam (FIG. 3 a) or a folded beamtype (having an “S-shape,” FIG. 3 b).

Springs 3 a, 3 b are anchored to a substrate of the sensor (not shown)via respective spring anchoring regions 4 a, 4 b (e.g., pillarsextending up to and connected to the substrate); springs thus extendfrom a respective projection 2 a, 2 b of the proof mass 2 to arespective spring anchoring region 4 a, 4 b. In the case of foldedsprings, the springs have a first longitudinal arm connected to therespective spring anchoring region 4 a, 4 b, and a second longitudinalarm (connected to the first longitudinal arm via connecting longitudinaland vertical arms) to the proof mass 2, see FIG. 3 b.

The resonating part of the sensing structure is constituted by two verythin resonant beams 5, 6, that extend laterally with respect to theproof mass 2 (adjacent to sides thereof that do not face springs 3 a, 3b); in the shown embodiment, resonant beams 5, 6 extend longitudinallyalong the axis A, parallel to reference axis y, laterally to the proofmass with respect to reference axis x.

In particular, resonant beams 5, 6 are attached to the substrate at afirst one of their ends, at corresponding beam anchoring regions 7 a, 7b, and are attached to a respective spring 3 a, 3 b at the second one oftheir ends. The position of the connection point of the resonant beamswith the respective springs is denoted with c in FIGS. 3 a and 3 b andthus extends from the respective spring 3 a, 3 b to the respective beamanchoring region 7 a, 7 b. The resonant beams thus form, with theportion extending from the connection point c to the respective springanchoring region 4 a, 4 b of the respective springs, a sort of“L-shaped” resonant structure. As it will be explained in detail in thefollowing, the position of the connection point c, of the resonant beamswith the respective springs, with respect to the position of the springanchoring regions, is a factor determining the electricalcharacteristics of the resonating sensing structure (in particular, interms of the amplification of the axial force). In more detail, theresonant beams 5, 6 are attached to the respective springs at theirfirst longitudinal arm (which is also connected to the related springanchoring region).

Driving and sensing of the resonant beams 5, 6 is achieved through twoparallel electrodes 10, 11, that are fixed to the substrate (in a waynot shown in detail) and extend, in pairs, parallel to the respectiveresonant beams, facing opposite sides thereof.

For zero external acceleration the resonators have the same nominalfrequency f₀. When an external acceleration a is applied along axis A,as shown in FIG. 3 c, one resonator (e.g., the one constituted byresonant beam 5) is subject to tension and the other resonator (e.g.,the one constituted by resonant beam 6) is subject to a compression ofthe same magnitude N, N1=−N2 (i.e., N1=+N, N2=−N), as shown in FIG. 3 c(where the deformations of the springs are shown and the resulting newposition of the proof mass 2 is depicted). Accordingly, the frequency f1of the first resonator increases, while the frequency f2 of the secondresonator decreases by substantially the same amount.

Combining the output electrical signals from the two resonators, andusing the above discussed equations (1) and (2) linearized around f₀, itis possible to obtain the following frequency difference:

$\begin{matrix}{{{f_{1} - f_{2}} \cong {f_{0}\left( {1 + {\alpha\frac{{NL}^{2}}{2{EI}}} - 1 + {\alpha\frac{{NL}^{2}}{2{EI}}}} \right)}} = {f_{0}\alpha\frac{{NL}^{2}}{EI}}} & (3)\end{matrix}$

As it is apparent or may be readily shown, the presence of tworesonators undergoing opposite axial forces provides several advantages:

the sensitivity of acceleration detection can be doubled by measuringthe difference between the frequency of the two resonators instead ofthe variation of frequency of a single resonator (the acceleration beingproportional to the frequency difference);

the linearity of the system is improved, i.e., the accelerometerresponse can be linearized in a wider range of accelerations

the skew-symmetric geometry is less sensitive to spurious effects ofthermal loading, since an inelastic effect causing pre-stress in theresonators is cancelled when considering the difference between thefrequencies.

The sensitivity of the accelerometer, defined as the resonator frequencyvariation produced by an acceleration of 1 g, increases with thedimension of the proof mass 2 but also depends on the position of theresonating beams 5, 6 with respect to the anchor points of the springs 3a, 3 b. In order to reduce the device size while keeping a highsensitivity, this position may be advantageously optimized by means ofan analytical approach.

In this respect, FIG. 4 shows the sensitivity of the accelerometer,derived from equation (3), as a function of the position of theconnection points c connecting the resonant beams 5, 6 with therespective springs 3 a, 3 b expressed as a coordinate along the springextension (the origin corresponding to the position of the springanchoring regions 4 a, 4 b), normalized with respect to the springlength L (see FIGS. 3 a, 3 b).

It may be noted that an optimal position for the connection point c maybe found, very close to the position of the spring anchoring region ofthe spring, at about 1/60 of its length L. For example, the connectionpoint c may be formed to be in the range of 1% and 2% of the springlength, L. The different curves correspond to different values of theinertial mass of the proof mass 2.

The above equations (2), (3) can be used for axially constrained beamsif transverse oscillations can be considered small with respect to abeam's height. This hypothesis, which is often reasonable for structuralproblems at the macroscale level, may in general not be valid formicrostructures as those of MEMS resonators. In this case nonlinearitiesof the dynamic response of the resonators alone (for zero externalacceleration) due to their very small cross section and hence theirsmall flexural stiffness have also to be considered. However, thepeculiar geometry proposed, schematically shown in FIG. 5 b (that may bedenoted as “L-shaped” resonator) as opposed to the traditional geometryshown in FIG. 5 a (standard “I-shaped” doubly clamped resonator), allowsto considerably lower the axial force due to the second order effects inthe resonator.

A coupled electromechanical analysis has been performed in order tocompute the axial force induced in the resonating beam for differentlevels of the voltage applied to the corresponding electrode, with thetwo geometric configurations. Both the beam and the dielectric mediumbetween the electrodes should be discretized to compute the electricfield and the corresponding mechanical response of the resonator. Sincelarge displacements are considered, the electrostatic problem should besolved on a varying domain, considering the deformed mesh (see forfurther details C. Comi “On geometrical effects in micro-resonators”Latin American Journal of Solids and Structures, 6, 73-87, 2009).

FIG. 6 shows the computed axial force versus maximum deflection for thetwo configurations. While for the “I-shaped” resonator, second ordereffects always induce a tensile axial force, in the case of “L-shaped”resonator the axial force is initially of compression due to the firstorder term, then, for high values of oscillation amplitude, thenonlinear tension term becomes predominant. In all the considered rangeof displacement the axial force turns out to be significantly smallerthan the one obtained for the “I-shaped” resonator and there is a valueof the oscillation for which the axial force vanishes. This feature isvery important since it turns to be possible to tune the actuationvoltage amplitude in such a way that the axial force vanishes and theoscillator response can be effectively considered as linear (zero axialforce for zero external acceleration).

From the above quasi-static electromechanical analyses it is alsopossible to compute the capacitance variation between the resonant beams5, 6 and the respective sensing electrodes 10, 11 for increasing voltageapplied to the excitation electrode. The results are shown in FIG. 7 fortwo different resonator lengths. The voltage which can be applied islimited by the pull-in value corresponding to the vertical growth in thecapacitance variation.

The dynamic three-dimensional response of the whole accelerometerstructure has also been studied by a finite element analysis. The firstmodal shape corresponds to the in-plane (xy) translation of the proofmass 2 in the sensing direction (axis A), as shown in FIG. 3 c. For thedevice shown in FIG. 8 a, the frequency of this mode is 688 Hz. Thesecond mode is an out-of-plane translation of the proof mass 2 and thesecond frequency is about 10 kHz, well separated from the first one. Themodes corresponding to the oscillation of the resonators (i.e. the 5thand 6th modes) have frequencies higher than 50 kHz.

The above discussed sensing structure for the MEMS resonantaccelerometer may be produced with known surface micro-machiningprocesses, for example using the so-called ThELMA (Thick Epipoly Layerfor Microactuators and Accelerometers) process, which has been developedby the present Applicant, to realize in-silicon inertial sensors andactuators.

The Thelma process allows the realization of suspended structures with arelatively large thickness (15 μm) anchored to the substrate throughvery compliant parts (springs) and thus capable of moving on a planeparallel to the underlying silicon substrate (the above discussed xyplane), such as the accelerometer structure previously described.

The process includes several manufacturing steps that include forming asubstrate thermal oxidation, depositing and patterning horizontalinterconnections, depositing and patterning a sacrificial layer, formingan epitaxial growth of a structural layer (15 μm thick polysilicon),patterning the structural layer by trench etching, and removing thesacrificial layer (oxide); and depositing contact metallization.

A manufactured accelerometer structure 30 is shown in an SEM image inFIG. 8 a. The very thin resonant beams 5, 6 are visible on the right andthe left of the proof mass 2, extending in the vertical direction (axisA) between the sensing and excitation electrodes 10, 11. Thesquare-shaped (in plane view) proof mass 2 has holes to allow thecomplete oxide removal below it (removal of the sacrificial region).

FIG. 8 b shows an enlarged detailed view of the anchoring region 4 a ofthe horizontal spring 3 a and the connection point c with the verticalresonating beam 5. The upper portions of the associated electrodes 10,11 are also visible.

From what has been described and illustrated previously, the advantagesthat the resonant accelerometer according to the disclosure enables areevident.

In particular, it is again underlined that the proposed MEMS sensingstructure allows to obtain high sensitivity values, with small overalldimensions (even lower than those of known capacitive accelerometers).

Moreover, the proposed structure allows reduction of the effects ofspurious axial forces on the resonant beams.

The optimized proposed design allows production of a very smallaccelerometer (for example, a proof mass of 400 μm×400 μm) with a highsensitivity (of about 450 Hz/g). If a proof mass with higher size isused (e.g., 700 μm×700 μm), an even higher sensitivity of 2 kHz/g may beobtained.

Moreover, the above high sensitivity values are obtained with lowresonant quality factors (Q), having values around 200 (considerablylower than known resonant sensing structures).

Finally, it is clear that modifications and variations may be made towhat has been described and illustrated herein, without therebydeparting from the scope of the present disclosure.

These and other changes can be made to the embodiments in light of theabove-detailed description. In general, in the following claims, theterms used should not be construed to limit the claims to the specificembodiments disclosed in the specification and the claims, but should beconstrued to include all possible embodiments along with the full scopeof equivalents to which such claims are entitled. Accordingly, theclaims are not limited by the disclosure.

The invention claimed is:
 1. A MEMS resonant accelerometer, comprising:a substrate; a proof mass; a first anchoring region coupled to thesubstrate; a first elastic element coupled to the first anchoring regionand the proof mass, the first elastic element configured to allowmovement along a sensing axis in response to an external acceleration; asecond anchoring region coupled to the substrate; a second elasticelement coupled to the second anchoring region and to the proof mass,the second elastic element configured to allow movement along thesensing axis in response to the external acceleration; a third anchoringregion coupled to the substrate; a first resonating element having aresonant frequency and a longitudinal extension along the sensing axiswith a first end and a second end, the first end coupled to the thirdanchoring region and the second end coupled to a first coupling point onthe first elastic element, the first resonating element being coupled tothe proof mass via the first elastic element, and configured to generatea first variation of the resonant frequency in response to a first axialstress when the proof mass moves along the sensing axis; a fourthanchoring region coupled to the substrate; and a second resonatingelement having the resonant frequency and a longitudinal extension alongthe sensing axis with a respective first end and a respective secondend, the respective first end coupled to the fourth anchoring region andthe respective second end coupled to a second coupling point on thesecond elastic element, the second resonating element being coupled tothe proof mass via the second elastic element, and being configured togenerate a second variation of the resonant frequency in response to asecond axial stress when the proof mass moves along the sensing axis,the second axial stress being substantially opposite to the first axialstress, and the second variation being substantially opposite to thefirst variation.
 2. The resonant accelerometer according to claim 1,wherein the first axial stress is a tensile stress and the second axialstress is a compression stress, the tensile and compression stresseshaving a same magnitude.
 3. The resonant accelerometer according toclaim 1, wherein the first and second coupling points are located alongthe first and second elastic elements, respectively, at a distance fromthe first and second anchoring regions, respectively, the distance is inthe range of 1/100 and 2/100 of a length of the first and second elasticelements.
 4. The resonant accelerometer according to claim 1, whereinthe first and second coupling points are located along the first andsecond elastic elements, respectively, at a distance from the first andsecond anchoring regions that is substantially equal to 1/60 of a lengthof the first and second elastic elements.
 5. The resonant accelerometeraccording to claim 1, wherein the proof mass has a first and a secondprojecting portion, and the first and second elastic elements extendbetween the first and second projecting portions and the first andsecond anchoring regions, respectively.
 6. The resonant accelerometeraccording to claim 5, wherein the first and second projecting portionsand the first and second anchoring regions are positioned so that thefirst and second elastic elements are substantially parallel to eachother and are transverse to the sensing axis and wherein the first andsecond resonating elements extend transverse to the first and secondelastic elements at opposite sides of the proof mass.
 7. The resonantaccelerometer according to claim 1, wherein the first and secondresonating elements have a longitudinal extension that is substantiallyparallel to the sensing axis.
 8. The resonant accelerometer according toclaim 5, wherein the proof mass has a substantially square shape in planview, and the first and second projecting portions extend fromdiagonally opposite corners thereof.
 9. An electronic device,comprising: a microprocessor; a detection circuit coupled to themicroprocessor; and a MEMS resonant accelerometer coupled to thedetection circuit, the accelerometer comprising: a substrate; a proofmass; a first, a second, a third, and a fourth anchoring region coupledto the substrate; a first elastic element coupled to the first anchoringregion and the proof mass; a second elastic element coupled to thesecond anchoring region and to the proof mass, the first and secondelastic elements configured to allow movement along a sensing axis inresponse to an external acceleration; a first resonating element havinga resonant frequency and a longitudinal extension along the sensing axiswith a first end and a second end, the first end coupled to the thirdanchoring region and the second end coupled to a first coupling point onthe first elastic element, the first resonating element being coupled tothe proof mass via the first elastic element, and configured to generatea first variation of the resonant frequency in response to a first axialstress when the proof mass moves along the sensing axis; and a secondresonating element having the resonant frequency and a longitudinalextension along the sensing axis with a respective first end and arespective second end, the respective first end coupled to the fourthanchoring region and the respective second end coupled to a secondcoupling point on the second elastic element, the second resonatingelement being coupled to the proof mass via the second elastic element,and configured to generate a second variation of the resonant frequencyin response to a second axial stress when the proof mass moves along thesensing axis, the second axial stress being substantially opposite tothe first axial stress, and the second variation being substantiallyopposite to the first variation; and wherein the detection circuit isconfigured to detect a value of the external acceleration as a functionof a difference between the first and second variations of the resonantfrequency.
 10. A MEMS resonant accelerometer, comprising: a substrate; aproof mass; a first anchoring region coupled to the substrate; a firstelastic element coupled to the first anchoring region and the proofmass, the first elastic element configured to allow movement along asensing axis in response to an external acceleration; a second anchoringregion coupled to the substrate; a second elastic element coupled to thesecond anchoring region and to the proof mass, the second elasticelement configured to allow movement along the sensing axis in responseto the external acceleration; a first resonating element having aresonant frequency and a longitudinal extension along the sensing axis,the first resonating element being coupled to the proof mass via thefirst elastic element, and configured to generate a first variation ofthe resonant frequency in response to a first axial stress when theproof mass moves along the sensing axis; and a second resonating elementhaving the resonant frequency and a longitudinal extension along thesensing axis, the second resonating element being coupled to the proofmass via the second elastic element, and being configured to generate asecond variation of the resonant frequency in response to a second axialstress when the proof mass moves along the sensing axis, the secondaxial stress being substantially opposite to the first axial stress, andthe second variation being substantially opposite to the firstvariation.
 11. The electronic device according to claim 10, wherein thelongitudinal extension of the first resonating element has a first endand a second end, the first end is coupled to a third anchoring regionon the substrate and the second end is coupled to a first coupling pointon the first elastic element; and the longitudinal extension of thesecond resonating element has a third end and a fourth end, the thirdend is coupled to a fourth anchoring region on the substrate and thefourth end is coupled to a second coupling point on the second elasticelement.
 12. The resonant accelerometer according to claim 11, whereinthe first and second coupling points are located along the first andsecond elastic elements, respectively, at a close distance from thefirst and second anchoring regions, respectively.
 13. A MEMS resonantaccelerometer, comprising: a substrate; a proof mass, having a first anda second projecting portion; a first anchoring region coupled to thesubstrate; a first elastic element coupled to the first anchoring regionand the first projecting portion of the proof mass, the first elasticelement configured to allow movement along a sensing axis in response toan external acceleration; a second anchoring region coupled to thesubstrate; a second elastic element coupled to the second anchoringregion and to the second projecting portion of the proof mass, thesecond elastic element configured to allow movement along the sensingaxis in response to the external acceleration, wherein the first andsecond projecting portions and the first and second anchoring regionsare positioned so that the first and second elastic elements aresubstantially parallel to each other and are transverse to the sensingaxis; a first resonating element having a resonant frequency and alongitudinal extension along the sensing axis, the first resonatingelement being coupled to the proof mass via the first elastic element,and configured to generate a first variation of the resonant frequencyin response to a first axial stress when the proof mass moves along thesensing axis; a second resonating element having the resonant frequencyand a longitudinal extension along the sensing axis, the secondresonating element being coupled to the proof mass via the secondelastic element, and being configured to generate a second variation ofthe resonant frequency in response to a second axial stress when theproof mass moves along the sensing axis, the second axial stress beingsubstantially opposite to the first axial stress, and the secondvariation being substantially opposite to the first variation; andwherein the first and second resonating elements extend transverse tothe first and second elastic elements at opposite sides of the proofmass.
 14. A method, comprising: providing a MEMS accelerometer having aproof mass coupled to a first and a second anchoring region via a firstand a second elastic flexible element, respectively, the first andsecond elastic flexible elements configured to allow the proof mass tomove along a sensing axis in response to an external acceleration,wherein the proof mass is coupled to a first and second resonatingelement via the first and second elastic flexible elements,respectively; activating the first and second resonating element toresonate at a resonant frequency; detecting a first variation of theresonant frequency generated by the first resonating element in responseto a first axial stress when the proof mass moves along the sensingaxis; detecting a second variation of the resonant frequency generatedby the second resonating element in response to a second axial stresswhen the proof mass moves along the sensing axis; and determining avalue of the external acceleration as a function of a difference betweenthe first and second variations.
 15. The method of claim 14 wherein thesecond axial stress is substantially opposite to the first axial stressand the second variation is substantially opposite to the firstvariation.
 16. The method of claim 14 wherein the first axial stress isa tensile stress and the second axial stress is a compression stress,the tensile and compression stresses having a same magnitude.
 17. Themethod of claim 14 wherein the first resonating element is coupledbetween a third anchoring region and a first connection point on thefirst elastic flexible element and the second resonating element iscoupled between a fourth anchoring region and a second connection pointon the second elastic flexible element.
 18. The device of claim 9wherein the first and second resonating elements extend transverse tothe first and second elastic flexible elements along opposite sides ofthe proof mass.
 19. The device of claim 9 wherein the first couplingpoint is spaced from the first anchoring region by a first distance andthe first coupling point is spaced from the proof mass by a seconddistance that is greater than the first distance.
 20. The device ofclaim 19 wherein the second coupling point is spaced from the secondanchoring region by the first distance and the second coupling point isspaced from the proof mass by the second distance.
 21. The accelerometerof claim 13 wherein the first resonating element is coupled between athird anchoring region and a first connection point on the first elasticflexible element and the second resonating element is coupled between afourth anchoring region and a second connection point on the secondelastic flexible element.
 22. The accelerometer of claim 13 wherein thefirst coupling point is spaced from the first anchoring region by afirst distance and the first coupling point is spaced from the proofmass by a second distance that is greater than the first distance. 23.The accelerometer of claim 22 wherein the second coupling point isspaced from the second anchoring region by the first distance and thesecond coupling point is spaced from the proof mass by the seconddistance.
 24. A device, comprising: a substrate; a proof mass; a firstanchoring region; a first elastic flexible element coupled to the firstanchoring region and the proof mass; a second anchoring region; a secondelastic flexible element coupled to the second anchoring region and tothe proof mass; a third anchoring region; a first resonating elementhaving a longitudinal extension in a first direction, the firstresonating element being coupled to the proof mass via the first elasticflexible element; a fourth anchoring region; and a second resonatingelement having a longitudinal extension in the first direction, thesecond resonating element being coupled to the proof mass via the secondelastic flexible element.
 25. The device of claim 24 wherein the firstelastic flexible element configured to allow movement along a sensingaxis in response to an external acceleration and the second elasticflexible element configured to allow movement along the sensing axis inresponse to the external acceleration.
 26. The device of claim 25wherein the first resonating element is configured to generate a firstvariation of the resonant frequency in response to a first axial stresswhen the proof mass moves along the sensing axis and the secondresonating element is configured to generate a second variation of theresonant frequency in response to a second axial stress when the proofmass moves along the sensing axis, the second axial stress beingsubstantially opposite to the first axial stress, and the secondvariation being substantially opposite to the first variation.
 27. Thedevice of claim 24 wherein the first resonating element has a resonantfrequency and the second resonating element has the same resonantfrequency.
 28. The device of claim 24 wherein the first resonatingelement is coupled between the third anchoring region and a firstcoupling point on the first elastic flexible element and the secondresonating element is coupled between the fourth anchoring region and asecond coupling point on the second elastic flexible element.
 29. Thedevice of claim 25 wherein the first and second elastic flexibleelements have a longitudinal extension in a second direction that istransverse to the first direction.
 30. The device of claim 29 whereinthe first direction is along the sensing axis.